A350541 Twin primes x, represented by their average, such that x is the first and x+18 the last of three successive twins.

12, 180, 810, 5640, 9420, 18042, 62970, 88800, 97842, 109830, 165702, 284730, 392262, 452520, 626610, 663570, 663582, 855720, 983430, 1002342, 1003350, 1068702, 1146780, 1155612, 1322160, 1329702, 1592862, 1678752, 1718862, 1748472, 2116560, 2144490

Offset

1,1

Comments

Subsequence of A014574. For x>6, d=18 is the least possible difference between the least and the greatest of three twins. For d=12, one of the six terms 6*k+-1, 6*k+6+-1,6*k+12+-1 would be divisible by 5. Therefore, d>12, except for x=6.

The distribution of 35314 terms < 10^11 is in accordance with the k-tuple conjecture, see links "k-tuple conjecture" and "Test of the k-tuple conjecture".

Generalizations:

Twin primes x such that x is the first and x+d the last of m successive twins.

m d

1 0 A014574(n) twin primes

2 6 A173037(n)-3

3 12 Only one quadruple: (6,12,18,30)

3 18 Current sequence

4 24 Only one quintuple: (6,12,18,30,42)

4 30 A350542

5 36 See A350543

5 42 See A350543

5 48 A350543

Links

Table of n, a(n) for n=1..32.

Gerhard Kirchner, Test of the k-tuple conjecture

Eric Weisstein's World of Mathematics, k-tuple conjecture.

Example

Triples of twins Example   6-tuple of primes
(x,x+ 6,x+18)     x= 12   (11,13,17,19,29,31)
(x,x+12,x+18)     x=180   (179,181,191,193,197,199)

Mathematica

Select[Prime@Range[4, 160000], Count[Range[#, #+18], _?(PrimeQ@#&&PrimeQ[#+2]&)]==3&]+1 (* Giorgos Kalogeropoulos, Jan 07 2022 *)

Prog

(Maxima)
block(twin:[], n:0, p1:11, j2:1, nmax: 3,
/*returns nmax terms*/
m:3, d:18, w: makelist(-d, i, 1, m),
while n<nmax  do(
p2: next_prime(p1), if p2-p1=2 then(
  k:p1+1, j1:j2, j2:1+ mod(j2, m), w[j1]:k,
  if w[j1]-w[j2]=d then(n:n+1, twin: append(twin, [k-d]))),
p1:p2), twin);

Crossrefs

Cf. A014574, A173037, A350542, A350543.

Sequence in context: A227714 A358951 A363152 * A130550 A073975 A069685

Adjacent sequences: A350538 A350539 A350540 * A350542 A350543 A350544

Keyword

nonn

Author

Gerhard Kirchner, Jan 06 2022

Status

approved